Whittaker category for the Lie algebra of polynomial vector fields

Abstract

For any positive integer n, let An=C[t1,…,tn], Wn=Der(An) and Δn=Span{∂∂t1,…,∂∂tn}. Then (Wn,Δn) is a Whittaker pair. A Wn-module M is called a Whittaker module if the action of Δn on M is locally finite. We show that each block ΩaW˜ of the category of (An,Wn)-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over Ln, where Ln is the Lie subalgebra of Wn consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker Wn-modules with finite dimensional Whittaker vector spaces using gln-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ΩaW.